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INVARIANT RINGS OF MODULAR

P -GROUPS

a thesis

submitted to the department of mathematics

and the graduate school of engineering and science

of bilkent university

in partial fulfillment of the requirements

for the degree of

master of science

By

Ceren Co¸skun Toper

January, 2013

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I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science.

Assoc. Prof. Dr. M¨ufit Sezer(Advisor)

I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science.

Assist. Prof. Dr. ¨Ozg¨un ¨Unl¨u

I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Master of Science.

Prof. Dr. Ay¸se C¸ i˘gdem ¨Ozcan

Approved for the Graduate School of Engineering and Science:

Prof. Dr. Levent Onural Director of the Graduate School

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ABSTRACT

INVARIANT RINGS OF MODULAR P -GROUPS

Ceren Co¸skun Toper M.S. in Mathematics

Supervisor: Assoc. Prof. Dr. M¨ufit Sezer January, 2013

We consider a finite group acting as linear substitutions on a polynomial ring and study the corresponding ring of invariants. Computing the invariant ring and finding its ring theoretical properties is a classical problem. We focus on the modular case where the characteristic of the field divides the order of the group. We review invariants of basic modular actions and give explicit descriptions of invariants of small dimensional actions. We also discuss a recent algorithm that computes the invariant ring of a modular p-group up to a localization and we apply this algorithm to invariants of indecomposable representations of a cyclic group of prime order.

Keywords: modular group actions, invariants, polarization, generating sets, lo-calization.

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¨

OZET

MOD ¨

ULER P -GRUPLARIN ˙INVARYANT HALKALARI

Ceren Co¸skun Toper Matematik, Y¨uksek Lisans Tez Y¨oneticisi: Do¸c. Dr. M¨ufit Sezer

Ocak, 2013

Sonlu bir grubun bir polinom halkasına lineer bir etkimesini ele alıp bu etkimeye kar¸sılık gelen invaryant halkayı ¸calı¸sıyoruz. ˙Invaryant halkasının hesaplanması ve cebirsel ¨ozelliklerinin bulunması klasik bir problemdir. Biz cismin karakter-isti˘ginin gruptaki eleman sayısını b¨old¨u˘g¨u mod¨uler duruma odaklanıyoruz. Temel mod¨uler etkimelerin invaryantlarını g¨ozden ge¸cirip k¨u¸c¨uk boyutlu bazı etkimelerin invaryantlarının ¨urete¸clerini a¸cık bir ¸sekilde veriyoruz. Mod¨uler bir p-grubun in-varyantlarını i¸ceren bir cismi hesaplayan, yakın zamanda bulunmu¸s bir algorit-mayı ¸calı¸sıp bu algoritalgorit-mayı asal mertebeli devirli grupların par¸calanamaz temsil-lerine uyguluyoruz.

Anahtar s¨ozc¨ukler : mod¨uler grup etkimeleri, invaryantlar, polarizasyon, ¨uretici k¨umeler, lokalizasyon .

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Acknowledgement

I would like to express my sincerest gratitude to my supervisor, Assoc. Prof. Dr. M¨ufit Sezer who has supported me throughout my thesis with his guide-ness, encouragement and valuable suggestions. One simply could not wish for a friendlier or better supervisor.

I would like to thank Assist. Prof. Dr. ¨Ozg¨un ¨Unl¨u and Prof. Dr. Ay¸se C¸ i˘gdem ¨Ozcan for reading this thesis.

I am grateful to my family members for their love and their support in every stage of my life.

I would like to express my gratitude to my dear husband T¨urkay Toper for his never-ending patience, continual support and love.

I also thank all my friends who have supported me in many ways during the creation period of this thesis.

The work that form the content of the thesis is supported financially by T ¨UB˙ITAK through the graduate fellowship program, namely “T ¨UB˙ITAK-B˙IDEB 2210- Yurt ˙I¸ci Y¨uksek Lisans Programı”. I am grateful to the council for their kind support.

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Contents

1 Introduction 1

2 Modular Invariants of Cyclic Group of Prime Order 4

2.1 Preliminaries . . . 4

2.1.1 Some Basic Invariants . . . 7

2.1.2 Modular Actions of Cp . . . 7

2.2 Invariants of V2 and Its Vector Sums . . . 9

2.2.1 Example: Cp Represented on a 2 Dimensional Vector Space in Characteristic p . . . 9

2.2.2 Further Example: Cp Represented on 2V2 in Characteristic p 12 2.2.3 The Vector Invariants of V2 . . . 15

2.3 Polarization . . . 17

3 Localized Invariant Rings of p-Groups 21 3.1 General Algorithm . . . 22

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CONTENTS vii

3.2.1 Construction of Invariants . . . 26

3.2.2 Localized Invariants of Indecomposable Representations of Cp . . . 28

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Chapter 1

Introduction

Invariant theory tries to determine whether a mathematical object can be ob-tained from some other object by the action of some group. One way to answer this question is to construct “invariant polynomials”. These are polynomial func-tions from the class of objects to some field which take the same value on any two objects which are related by an element of the group. Thus if we can find any invariant which takes different values on two objects, then these two objects cannot be related by an element of the group. Ideally, one would like to find enough invariants to distinguish as many orbits as possible. This means we want to find a (finite) set of invariants f1, f2, . . . , fr with the property that if two

ob-jects are not related by the group action then at least one of these r invariants takes different values on the two objects in question.

This fascinating field was brought to life at the beginning of the last century. One of the central results in this area is the theorem about the finiteness of the number of fundamental invariants which is proved by Hilbert at the end of the nineteenth century. The celebrated theorems of Hilbert, like the Basis Theorem, the Syzygy Theorem, and Nullstellensatz were all born as lemmas for proving important theorems in invariant theory. Generally speaking, the invariant theory has applications in commutative algebra, topology, geometry and homological algebra, and on the other direction, these fields are sources that provide powerful tools to attack problems in invariant theory.

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A central problem in invariant theory is to determine the invariant ring by finding a generating set explicitly. Knowing the degrees of the generators is very critical for this purpose because it reduces the problem to a linear algebra problem in a finite dimensional vector space. In the first quarter of the twentieth century, Noether made a big contribution to these problems with obtaining a bound on degrees of the generators. Specifically, she proved that if G is finite and the characteristic of the field does not divide |G| (i.e. in the non-modular case), then the ring of invariants for G is always generated by invariants of degree at most |G|. This bound is called the Noether bound. However, Noether originally proved this over fields of characteristic zero. To extend this bound to all characteristics not dividing the order of the group could be achieved almost a hundred years later, [1], [2].

Many questions which are well understood in the non-modular case are still open in the modular case(i.e. when the characteristic of field divides |G|). Mod-ular invariant theory is a recent trend which has emerged as an active research area in the last decade or so. Many tools and techniques that working the non-modular case do not carry over to the non-modular situations and non-modular invariant rings tend to be more complicated. Generally speaking, the degrees of the gen-erators grow large and the invariant ring moves away from being a regular ring which means we get more and more relations among the generators. For instance, Noether bound is not true in general in the modular case. In fact, we know that one can not get a bound that depends only on the group order. Due to this complications, finding an explicit generating set for the invariants of a modular group is a difficult task. Even in the simplest modular case of a cyclic group of prime order explicit generating sets are known for a handful of cases only and we are still a very long way from being able to write down algebra generators in the general case. We direct the reader to [3], [4], [5], [6], [7], [8], [9] for some major results on this matter.

In this thesis, we restrict to the modular case and start by describing the modular actions of a cyclic group of prime order. We review basic properties of its invariants and give explicit descriptions of invariants of any vector copies of

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the two dimensional indecomposable representations. Then we study an algo-rithm due to Chuai and Campbell [10] that computes the invariants of a modular p-group up to a localization. This algorithm realize on computing invariant poly-nomials with minimal positive degree in certain variables and our background on invariants of cyclic groups of prime order allows us to apply this algorithm to these groups and we obtain a localized invariant subalgebra that contains the full invariant ring. This method has the potential to be very useful to study the invariants of non-cyclic abelian modular p-groups and this provides us a perspec-tive for future research. Another tool in constructing these critical polynomials is the polarization. This is a classical tool in invariant theory that goes back to Weyl and we also have a section that provides a background on polarization.

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Chapter 2

Modular Invariants of Cyclic

Group of Prime Order

2.1

Preliminaries

We consider a finite dimensional representation ρ of a group G over a field F, i.e., a group homomorphism

ρ : G → GL(V )

where V is a finite dimensional vector space over a field F.

The representation defines a left action of the group G on V . Given σ ∈ G and v ∈ V we write σ(v) for the vector ρ(σ)(v), the result of applying ρ(σ) to v.

We denote the set of vectors fixed (pointwise) by the group G by

VG = {v ∈ V | σ(v) = V, for all σ ∈ G}.

Now consider V∗, the vector space dual to V . This dual space V∗ consists of all linear functionals from V to F and is denoted by HomF(V, F). Recall that

x : V → F is said to be a linear functional if x(av + bw) = ax(v) + bx(w) for all v, w ∈ V , and all a, b ∈ F. Of course, we have dimF(V

) = dim F(V ).

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The action of G on V determined by the representation ρ naturally induces a left action of G on V∗ as follows. Let x ∈ V∗ be any linear functional on V and let σ ∈ G. Then σ(x) should be another linear functional on V . This new linear functional is defined by (σ(x))(v) := x(σ−1(v)). In this definition we use σ−1 instead of σ in order to obtain a left action (and not a right action) of G on V∗. This new representation of G is often referred to as the dual representation. We extend this action linearly to the entire vector space V∗. This gives a left linear action of G on V∗. We obtain a faithful action iff the representation of G is faithful.

Often we simultaneously consider actions on V and V∗ and so we note down how to pass from one action to the other.

Lemma 2.1.1. Suppose we have a fixed representation ρ : G → GL(V ) and consider also ρ∗ : G → GL(V∗). In general, for σ ∈ G the matrix representing ρ(σ) ∈ GL(V ) with respect to a fixed basis is the transpose inverse of the matrix representing ρ∗(σ) with respect to the dual basis.

So far we have an action of G on the vector space V given by a linear repre-sentation ρ : G → GL(V ) which induces an action of G on the dual space V∗. This action also naturally induces an action of G on all polynomial functions as we see next.

Let {x1, x2, . . . , xn} be a fixed basis of V∗. Then we denote by F[V ] the ring

of polynomials in n indeterminants x1, x2, . . . , xn with coefficients from a field F,

i.e., F[V ] = F[x1, . . . , xn].

For an exponent sequence I = (i1, . . . , in) consisting of non-negative integers,

we define the monomial

xI = x1i1. . . xnin.

We say that xI has degree i

1+ . . . + inand we denote the degree of xI by deg(xI)

or deg(I). An arbitrary polynomial f ∈ F[V ] can be written as a finite sum

f (x1, . . . , xn) =

X

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As usual, we say that a polynomial f = P ajxIj for aj ∈ F is homogeneous

of degree d if each of its monomials, xIj, is of degree d. We observe that F[V ]

is naturally graded by degree: we may write F[V ] = ⊕d≥0F[V ]d where F[V ]d

denotes the subspace of homogeneous polynomials of degree d (including the zero polynomial). We also observe that F[V ] is a graded algebra. This just means that each F[V ]dis a subspace and that if f ∈ F[V ]dand f0 ∈ F[V ]0dthen f f

0

∈ F[V ]d+d0.

We define the G-action on a monomial as

σ(x1i1. . . xnin) = (σ(x1))i1. . . (σ(xn))in

for σ ∈ G, i.e., we extend the action multiplicatively. To obtain an action of G on all polynomials, we also extend the action linearly:

σf = σ(XajxIj) =

X

ajσ(xIj).

Thus altogether we have

σf (v) = f (σ−1(v)), for all σ ∈ G, v ∈ V and f ∈ F[V ].

The main object of study in invariant theory is the collection of polynomial functions on V left fixed by all of G. We call a polynomial f ∈ F[V ] invariant under the group action of G if

σ(f ) = f, for all σ ∈ G.

We denote by F[V ]G⊆ F[V ] the subset of all invariant polynomials. That is

F[V ]G := {f ∈ F[V ] | σ(f ) = f for all σ ∈ G}.

Since the action is additive and multiplicative, sum of and product of invariant polynomials are also invariant. Thus the subset of all invariant polynomials is a subring. This ring is called the ring of invariants.

Since the group action preserves degree, if a polynomial is invariant under the group action of G then its homogeneous components are also invariant. This makes F[V ]G a graded subalgebra.

For a general reference for invariant theory we recommend [11], [12], [13], [14], [15], [16].

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2.1.1

Some Basic Invariants

We have general methods to construct invariants of finite groups as follows.

Let f ∈ F[V ]. Then the transf er or trace of f is defined as

T r(f ) = T rG(f ) =X

g∈G

g(f ).

Note that for all h ∈ G we have

hT rG(f ) =X g∈G h(gf ) =X g∈G (hg)f = X g0∈G g0(f ) = T rG(f ),

because if g ∈ G runs through all group elements, then hg (for a fixed h) does too. Therefore the image of the transfer is contained in the ring of invariants

T rG : F[V ] → F[V ]G. Similarly, the norm of f is defined by

N (f ) = NG(f ) = Y

g∈G

g(f ).

2.1.2

Modular Actions of C

p

Let G = Cp denote the cyclic group of order p. In this section, we study the

representation of Cp over a field F of characteristic p.

We consider a finite dimensional indecomposable Cp-module V . In

repre-sentation terminology, this means we have a finite dimensional indecomposable representation ρ : G → GL(V ). We fix a generator σ of Cp. Now since σp = 1,

every eigenvalue λ of σ must be a pth root of unity. Thus 0 = λp− 1 = (λ − 1)p

and so λ = 1 is the only pth root of unity in F. Since the only eigenvalue of σ lies in F, we may choose a basis {e1, e2, . . . , en} of V such that ρ(σ) is in (lower)

Jordan normal form with respect to this basis. If there are two or more Jordan blocks in this Jordan normal form these blocks yield a decomposition of V into a

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direct sum of smaller Cp-modules contradicting the indecomposability of V . Thus ρ(σ) =              1 0 0 . . . 0 0 1 1 0 . .. 0 0 0 1 1 . .. 0 0 .. . ... . .. ... ... ... 0 0 0 . .. 1 0 0 0 0 . . . 1 1             

Lemma 2.1.2. The matrix above has order pl if and only if pl−1 < n ≤ pl. In particular, this matrix has order p if and only if 1 < n ≤ p.

Definition 2.1.3. For each n with 1 ≤ n ≤ p we denote the indecomposable Cp-module of dimension n by Vn.

We note that σ(en) = enand σ(ei) = ei+ ei+1 for 1 ≤ i ≤ n − 1. We call such

a basis a triangular basis of Vn and we say that e1 is distinguished. Notice that

the Cp-module generated by e1 is all of Vn.

If the matrix ρ(σ) has dimension greater than pr then its order will be also

greater than pr and so greater than the order of the group which cannot be happen. Therefore, the matrix can have order at most pr. Also note that every dimension in the Jordan block corresponds to a indecomposable Cp-module. Thus

there can be at most pr many indecomposable C

p-modules.

The preceding discussion , combined with Lemma 2.1.2, proves the following lemma.

Lemma 2.1.4. Over any field of characteristic p, there are exactly pr inequivalent indecomposable representation of Cpr, one of dimension n for each n less than or

equal to pr. Furthermore, we have the following chain of inclusions:

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2.2

Invariants of V

2

and Its Vector Sums

2.2.1

Example: C

p

Represented on a 2 Dimensional

Vec-tor Space in Characteristic p

As a simple example of a modular group action, consider the vector space V2 of

dimension 2 over a field F of characteristic p > 0 with basis {e1, e2}.

Let Cp denote the cyclic group of order p generated by σ. Consider the matrix

τ = 1 0 1 1

!

inside GL(2, F) where F is a field of characteristic p. Using induction, it is easy to show that

τi = 1 0 i 1

!

Therefore, we obtain a representation ρ : Cp → GL(V2) given by the rule ρ(σi) =

τi. We have σ(e1) = τ (e1) = e1+ e2 and σ(e2) = τ (e2) = e2.

Let {x, y} be the basis for V2∗ dual to {e1, e2}. Then the action on the dual

basis is σ(x) = x and σ(y) = −x + y by Lemma 2.1.1.

We see immediately that the polynomial x is an invariant under this action. Moreover, since (y + x)p = yp+ xp,

σ(yp − xp−1y) = σ(y)p− σ(x)p−1σ(y)

= (y − x)p− xp−1(y − x) = yp− xp− xp−1y + xp

= yp− xp−1y

Therefore the polynomial N = yp − xp−1y is another example of an invariant.

Now, we will see that for this representation, these two invariants are the two most important invariants.

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Our aim is to show that the ring of Cp-invariants is the algebra generated by

the two invariants N and x.

Theorem 2.2.2. F[V2]Cp = F[x, N].

Before proving this theorem, we need to prove some preliminary results.

Lemma 2.2.3. Let f ∈ F[V2]. Then degy(σ(f )) = degy(f ).

Proof. Let degy(f ) = m and f = amym+ am−1ym−1+ . . . + a0

where ai ∈ F[x] and am 6= 0. Then

σ(f ) = σ(am)σ(y)m+ σ(am−1)σ(y)m−1+ . . . + σ(a0)

= am(y − x)m+ am−1(y − x)m−1+ . . . + a0

= amym+ terms of lower order in y

Thus degy(σ(f )) = m.

N is monic when considered as a polynomial in the variable y with coefficients from F[x], so we may divide any polynomial f ∈ F[x, y] by N to get f = qN + r where q, r ∈ F[x, y] are unique with degy(r) < p = degy(N ).

Lemma 2.2.4. If f ∈ F[V2]Gand f = qN +r with degy(r) < p, then q, r ∈ F[V2]G.

Proof. We note that it is enough to show that q and r are σ-invariant since σ generates Cp.We have qN + r = f = σ(f ) = σ(q)σ(N ) + σ(r) = σ(q)N + σ(r)

and since degy(σ(r)) = degy(r) < p (by the previous lemma), by the uniqueness of remainders and quotients we must have σ(r) = r and σ(q) = q.

Now we need a result concerning the partial differential operator ∂y∂. Lemma 2.2.5. If f ∈ F[x, y]G, then

∂y(f ) ∈ F[x, y] G.

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Proof. It is enough to show that σ(∂y∂(f )) = ∂y∂ (f ) = ∂y(σ(f )) since f ∈ F[x, y]G.

Therefore, we need to show that σ and ∂y∂ commute. Also, since both σ and ∂y∂ are F-linear maps, we need only to show that they commute on monomials:

σ( ∂ ∂y(x ayb) = σ(bxayb−1) = bxa(y − x)b−1 and ∂ ∂y(σ(x ayb)) = ∂ ∂y(x a(y − x)b) = bxa(y − x)b−1

Now we can give the proof of the Theorem 2.2.2.

Proof. We know that F[x, N] ⊆ F[V2]Cp. Thus it remains to prove that each

invariant is contained in F[x, N]. To prove this we use induction on degree in y.

Let f ∈ F[V2]Cp. If degy(f ) = 0, then f ∈ F[x] ⊆ F[x, N].

Assume that every invariant whose degree in y is less than d lies in F[x, N], and now suppose that degy(f ) = d. Let us divide f by N and get f = qN + r where m := degy(r) < p. We will show that m = 0. By way of contradiction, assume that m ≥ 1. By Lemma 2.2.4 and Lemma 2.2.5, q, r ∈ F[V2]Cp and if we apply

the partial differential operator ∂y∂ to r, we get another invariant polynomial. Let h be defined by

h := ∂

m−1(r)

∂ym−1 .

Then h = ay + b, where a is a non-zero scalar and b ∈ F[x]. But on the other side, h = σ(h) = σ(ay + b) = a(y − x) + b = ay + b − ax, so this contradiction shows that we must have m = 0. Therefore, f = qN + r where r ∈ F[x], q ∈ F[V2]Cp and degy(q) = d − p. By the induction hypothesis, q ∈ F[x, N] and

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2.2.2

Further Example: C

p

Represented on 2V

2

in

Char-acteristic p

Here we compute the ring of invariants of the group Cp on 2V2 = V2 ⊕ V2. But

first we give the following useful lemma.

Lemma 2.2.6. Suppose H is a normal subgroup of G with quotient group G/H. Let V be a representation of G. Then G/H acts naturally on VH and VG =

(VH)G/H.

We apply this lemma to a normal subgroup H of a group G acting on a coordinate ring F[V ]. Then we have F[V ]G= (F[V ]H)G/H.

Remark 2.2.7. Let G be any p-group for p a prime and let H be any maximal proper subgroup. Then H is normal in G necessarily of index p. Hence if G is generated by H and σ, we have G/H = Cp generated by ¯σ, the image of σ in

G/H.

Now we are considering the action of Cp determined by

σ =       1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1       We introduce σ1 =       1 0 0 0 −1 1 0 0 0 0 1 0 0 0 0 1       and σ2 =       1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1      

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Thus σ = σ2σ−11 .

Let {x1, y1, x2, y2} be the basis for the vector space (2V2)∗ dual to the standard

basis of 2V2. Thus σi(xj) = xj for 1 ≤ i, j ≤ 2, σi(yj) = yj for 1 ≤ j 6= i ≤ 2,

σ1(y1) = y1+ x1 and σ2(y2) = y2− x2. Define G to be the group generated by σ1

and σ2, so that G = Cp× Cp and H to be the group generated by σ = σ2σ1−1 so

that H = Cp. We want to compute F[2V2]H = F[2V2]Cp.

Given the example we computed in the section 2.2.1, it is easy to see that

F[2V2]G = F[V2]Cp⊗ F[V2]Cp = F[x1, N (y1), x2, N (y2)].

Now we consider the diagram

F(2V2)G ,→ F(2V2)H ,→ F(2V2)

↑ ↑ ↑

F[2V2]G ,→ F[2V2]H ,→ F[2V2]

where the vertical arrows are also inclusions. By Galois Theory, we have that the field F(2V2)H is an extension of the field F(2V2)G of degree p, that is, F(2V2)H as

vector space over F(2V2)G has dimension p. In order to exploit this property, we

need to find an element of F(2V2)H that lies outside of F(2V2)G.

It is easy to see that the element u = x1y2−x2y1is an invariant of least degree

in F[2V2]H outside F[2V2]H, and therefore F(2V2)H has basis

{1, u, u2, . . . , up−1}. We see that up = xp1N (y2) − xp2N (y1) + xp−11 x p−1 2 u. Theorem 2.2.8. F[2V2]Cp = F[x1, x2, N (y1), N (y2), u]. In fact, F[2V2]Cp = ⊕ p−1 i=0F[x1, x2, N (y1), N (y2)]ui.

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Proof. Since we noted above that F[2V2]G = F[x1, N (y1), x2, N (y2)], our aim is to

show that {1, u, u2, . . . , up−1} is a basis for F[2V

2]H as a module over F[2V2]G.

Since G is abelian, we have that H is normal in G and hence by Lemma 2.2.6

F[2V2]G = (F[2V2]H)G/H.

We note that the image of either σ1 or σ2 generates G/H = Cp.

We define

∆1 = σ1− Id

∆2 = σ2− Id and

∆ = σ − Id

Consider f ∈ F[2V2]H. Then we have σ(f ) = f and note that this implies

σ1(f ) = σ2(f ) and thus ∆1(f ) = ∆2(f ). In particular, f ∈ F[2V2]G if and only if

∆1(f ) = 0. Also, for f ∈ F[2V2]H, we have

σ(∆1(f )) = σ2σ1−1(σ1(f ) − f ) = σ2(f ) − σ(f ) = σ1(f ) − f = ∆1(f ).

Thus ∆1 : F[2V2]H → F[2V2]H.

Lemma 2.2.9. If f ∈ F[2V2]H, then ∆1(f ) = x1x2f0 for some f0 ∈ F[2V2]H.

Proof. For any f ∈ F[2V2], we write f =

Pd

l=0fly1l with fl ∈ F[x1, x2, y2] for 0 ≤

l ≤ d. Then by the action of σ1, we see that ∆1(f ) =

Pd

l=0fl(y1+x1)l−

Pd

l=0fly1l.

Thus ∆1(f ) = x1f0. Similarly, ∆2(f ) = x2f00. If f ∈ F[2V2]H, then σ1(f ) = σ2(f )

as we noted above, and so ∆1(f ) = ∆2(f ), that is, x1f0 = x2f00. But x1 and x2

are co-prime in F[2V2] and so ∆1(f ) = x1x2f000 for some f000 ∈ F[2V2]. Since both

∆1(f ) and x1x2 are H-invariant, we see that f000 ∈ F[2V2]H.

We now finish the proof of Theorem 2.2.8. Since ∆p1 = (σ1−Id)p = σ p

1−Id = 0

we see that ∆p1(f ) = 0 for all f ∈ F[2V2]. Thus given 0 6= f ∈ F[2V2]H there must

exist an l, 0 ≤ l < p with the property that 0 6= ∆l

1(f ) ∈ F[2V2]H and ∆l+11 (f ) = 0.

We claim that then f =Pl

m=0fmu

m for f

m∈ F[2V2]G. We proceed by induction

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For the general case, we write ∆1(f ) = x1x2f0 with f0 ∈ F[2V2]H and observe that f0 =Pl−1 m=0f 0 mum with all f 0

m ∈ F[2V2]G by induction. Now consider

∆l1(f + uf0/l) = ∆l1(f + 1 l l−1 X m=0 fm0 um+1) = ∆l−11 (∆1(f ) + 1 l l−1 X m=0 fm0 ∆1(um+1)) = ∆l−11 (x1x2f0+ 1 lf 0 l−1∆1(ul) + 1 l l−2 X m=0 fm0 ∆1(um+1)) = ∆l−11 ( l−1 X m=0 x1x2fm0 um+ 1 lf 0 l−1 l−1 X i=0 l i  ui(−x1x2)l−i + 1 l l−2 X m=0 fm0 ∆1(um+1)) = ∆l−11 ( l−2 X m=0 x1x2fm0 um+ 1 lf 0 l−1 l−2 X i=0 l i  ui(−x1x2)l−i + 1 l l−2 X m=0 fm0 ∆1(um+1)) = ∆l−11 ( l−2 X m=0 hmum)

where hm ∈ F[2V2]G for m = 1, 2, . . . , l − 2 and this final expression is equal

to zero since, as is easily verified, ∆t

1(us) = 0 whenever t > s. Therefore we

have ∆l1(f + uf0/l) = 0, and so f + uf0/l ∈ ⊕l−1m=0F[2V2]Gum by the induction

hypothesis. Thus f ∈ ⊕lm=0F[2V2]Gum which proves Theorem 2.2.8.

2.2.3

The Vector Invariants of V

2

Given a representation V of a group G and an integer m ≥ 2, a ring of invariants F[mV ]G is called a ring of vector invariants. A theorem providing an explicit description of F[mV ]Gfor all m ≥ 1 is called a first fundamental (or main) theorem for V . The following first fundamental theorem for V2 was conjectured by David

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Richman and proved by Campbell and Hughes, see [17]. Their proof is technical and uses a deep result about the rank of zero-one matrices in characteristic p.

Theorem 2.2.10. Let G = Cp = hσi act on V = mV2. Let {yi, xi} denote a

basis for the ith copy of V∗ 2 in V

where σ(y

i) = yi + xi and σ(xi) = xi. Thus

{x1, y1, x2, y2, . . . , xm, ym} is an upper triangular basis for V∗. Then the ring of

invariants F[mV2]G is generated by the following invariants:

1. xi for i = 1, 2, . . . , m. 2. NCp(y i) = yip− x p−1 i yi for i = 1, 2, . . . , m. 3. uij = xjyi− xiyj for 1 ≤ i < j ≤ m. 4. T rCp(ya1 1 y a2 2 . . . ymam) where 0 ≤ ai < p for i = 1, 2, . . . , m.

Remark 2.2.11. Shank and Wehlau [18] showed that if a1+a2+. . .+am ≤ 2(p−

1), then T rCp(ya1

1 y a2

2 . . . yamm) lies in the subalgebra generated by x1, x2, . . . , xm

and uij with 1 ≤ i < j ≤ m. Additionally, they also showed that if we exclude

invariants of this form, the remaining invariants minimally generate F[mV2]Cp.

The following example illustrates Theorem 2.2.10.

Example 2.2.12. If we take m = 3 and F a field of characteris-tic p = 3, then Theorem 2.2.10 tells us that F[3V2]C3 is generated by

x1, x2, x3, N (y1), N (y2), N (y3), u12, u13, u23 and some transfers.

It is straightforward to compute T rC3(y i) = 0 for i = 1, 2, 3; T rC3(y iyj) = −xixj for 1 ≤ i, j ≤ 3; T rC3(y2 iyj) = xiuji for 1 ≤ i 6= j ≤ 3; T rC3(y 1y2y3) = x1u23− x3u12; T rC3(y2 iy 2 j) = −uij − x2ix 2 j for 1 ≤ i < j ≤ 3; T rC3(y

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Thus we see that F[3V2]C3 is minimally generated by x1, x2, x3, N (y1), N (y2), N (y3), u12, u13, u23, T rC3(y12y 2 2y3), T rC3(y12y2y32), T rC3(y 1y22y 2 3) and T r C3(y2 1y 2 2y 2 3).

2.3

Polarization

We call the ring F[mV ]G a ring of vector invariants of G where mV = ⊕mV the

coordinate ring with the diagonal action of G.

Consider the map φ : mV → V given by φ(v1, v2, . . . , vm) = v1+ v2+ . . . + vm.

This map is GL(V )-equivariant where GL(V ) acts diagonally on mV .

This map naturally induces ring map φ∗ : F[V ] → F[mV ] given by (φ∗(f ))(v1, v2, . . . , vm) = f (φ(v1, v2, . . . , vm)) = f (v1+ v2+ . . . + vm).

We define the polarization map in terms of the variables as follows.

Let V be an n-dimensional vector space over F, with standard dual basis {x1, . . . , xn} and let {x11, x12, . . . , xij, . . . , xnm} be the dual basis for mV . We

define the polarization map by

P ol : F[V ]d→ F[mV ]d, xi 7→ xi1+ . . . + xim,

in other words we send the ith basis vector to the ith row sum of the matrix [xij]ij. We extend this map additively and multiplicatively and thus obtain a

homomorphism of F-algebras.

Let f ∈ F[V ]d. Using the Nm-grading on F[mV ] we have

φ∗(f ) = X

i1+i2+...im=d

f(i1,i2,...,im)

where each f(i1,i2,...,im)∈ F[mV ](i1,i2,...,im). These polynomials f(i1,i2,...,im)are called

partial polarizations of f and we write

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to denote the set of all such partial polarizations.

In order the compute individual polarizations, we take m indeterminates λ = (λ1, λ2, . . . , λm), and consider v = (v1, v2, . . . , vm) where each vi represents a

generic element of V . We write λv = λ1v1+ λ2v2+ . . . + λmvm, and then we have

f (λv) = f (φ(λ1v1, λ2v2, . . . , λmvm)) = φ∗(f )(λ1v1, λ2v2, . . . , λmvm) = X i1+i2+...im=d λi1 1 λ i2 2 . . . λ im mf(i1,i2,...,im)(v1, v2, . . . , vm) =X I λIfI(v)

with |I| = i1+ i2+ . . . + im = d where

fI ∈ F[mV ]I = F[V ]i1 ⊗ F[V ]i2 ⊗ . . . ⊗ F[V ]im ⊂ F[mV ]d.

As a special case, we may take m = d = deg(f ) and (i1, i2, . . . , im) =

(1, 1, . . . , 1) to get the full polarization of f denoted by P (f ) = f(1,1,...,1)= fmulti-linear∈ F[dV ].

Lemma 2.3.1. The mapping f 7→ f(i1,i2,...,im) is GL(V)-equivariant. In

particu-lar, if G is any subgroup of GL(V) and f ∈ F[V ]G, then P olm(f ) ⊂ F[mV ]G.

Proof. Let σ ∈ GL(V ). We need to show that (σf )I = σ(fI). The former is

defined by the equation

(σf )(λv) =X I λI(σf )I(v). But (σf )(λv) = f (λσ−1(v)) =X I λIfI(σ−1(v)). Therefore, (σf )I(v) = fI(σ−1(v)) = (σfI)(v).

In particular, if f is invariant then

(σf )I = fI = σ(fI)

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The following example illustrates polarization.

Example 2.3.2. Let K be a field of any characteristic. Consider the usual three dimensional permutation representation V of Σ3, the symmetric group on three

letters. Let {x, y, z} be a permutation basis for V∗. It is well known that if K has characteristic zero, then K[V ]Σ3 is the polynomial ring K[s

1, s2, s3] where

s1 = x + y + z, s2 = xy + xz + yz and s3 = xyz are elementary symmetric

polynomials. This result is also true when K has positive characteristic, even for characteristic 2 and 3.

Here we consider the ring of vector invariants K[2V ]Σ3. Weyl [19] proved that

the polarizations of the elementary symmetric functions f = s1, g = s2, h = s3

suffice to generate K[2V ]Σ3 if 6 = |Σ

3| is invertible in K. In fact, he proved that

if V is the usual permutation representation of Σn, then for any n and any m the

polarizations of the elementary symmetric polynomials s1, s2, . . . , sngenerate the

ring K[mV ]Σn provided only that n! is invertible in K. Here we have

X λIfI = f (λ1x1+ λ2x2, λ1y1+ λ2y2, λ1z1+ λ2z2) = λ1(x1+ y1+ z1) + λ2(x2+ y2+ z2). Thus P ol2(f ) = {f10, f01} where f10= x1+ y1+ z1, f01= x2+ y2+ z2. Similarly, X λIgI = g(λ1x1+ λ2x2, λ1y1+ λ2y2, λ1z1+ λ2z2) = λ21(x1y1+ x1z1+ y1z1) + λ1λ2(x1y2+ x1z2+ y1x2+ y1z2 + z1x2 + z1y2) + λ22(x2y2+ x2z2+ y2z2). Thus P ol2(g) = {g 20, g11, g02} where g20 = x1y1+ x1z1+ y1z1, g11 = x1y2+ x1z2+ y1x2+ y1z2+ z1x2+ z1y2, g02 = x2y2+ x2z2+ y2z2.

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Here g11 is the full polarization P (g). Finally, X λIhI = h(λ1x1+ λ2x2, λ1y1+ λ2y2, λ1z1+ λ2z2) = λ31(x1y1z1) + λ21λ2(x1y1z2+ x1y2z1+ x2y1z1) + λ1λ22(x1y2z2+ x2y1z2+ x2y2z1) + λ32x2y2z2. Hence P ol2(h) = {h 30, h21, h12, h03} where h30= x1y1z1, h21= x1y1z2 + x1y2z1+ x2y1z1, h12= x1y2z2 + x2y1z2+ x2y2z1, h03= x2y2z2.

Weyl’s result tells us that if the characteristic of K is neither 2 nor 3, then K[2V ]Σ3

is generated by the nine invariants

f10, f01, g20, g11, g02, h30, h21, h12, h03.

It turns out that these nine invariants also generate K[2V ]Σ3 if K has characteristic

2. The identity

3(x1y1z22+ y1z1x22+ x1z1y22) = f 2

10g02− f10f01g11+ f10h12+ g112

− 2f10h12+ f012g11− 4g20g02+ 2f01h21

shows how to express the invariant k := x1y1z22+ y1z1x22+ x1z1y22 in terms of the

polarized elementary symmetric functions when 3 is invertible. However, over a field of characteristic 3, this identity expresses an algebraic relation among the polarized elementary symmetric functions. In fact, over a field of characteristic 3, it is not possible to express k as a polynomial in the nine polarized elementary symmetric functions, and it turns out that the nine polarized elementary sym-metric functions together with the invariant k form a minimal generating set for K[2V ]Σ3 when K has characteristic 3.

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Chapter 3

Localized Invariant Rings of

p-Groups

In this chapter, we study an algorithm due to Chuai and Campbell [10] that computes the invariants of a modular p-group up to a localization.

Let F(V )G denote the quotient field of F[V ]G. It is easy to see that F(V )G is

also the G-invariant subfield of the quotient field F(V ). It is a famous question of Noether’s whether or not F(V )G is purely transcendental. The answer to this question is negative in general. However, if p > 0 and G is a p-group, then F(V )G

is purely transcendental [20].

If R denotes a commutative ring and b ∈ R, we denote by R(b) the localization

of R at the multiplicative set {1, b, b2, . . .}. In this chapter, we show that, for G

a p-group, a minimal generating set for F(V )G can be taken as homogeneous invariants from the invariant ring. If {a1, . . . , an} is such a set,we show that there

exists an element b ∈ F[a1, . . . , an], such that F[V ]G(b) = F[a1, . . . , an](b). This

generalizes the result of [21], which examines the case G = Cp, the cyclic group

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3.1

General Algorithm

Consider a representation V of a p-group G over a field F of characteristic p > 0. We may assume that G is a subgroup of the upper triangular group U (V ) = Un(F): any p-subgroup of GL(V ) is triangularizable.

Consequently, we can choose a basis {x1, . . . , xn} for V∗ such that (σ − 1)xi is

in the span of {x1, . . . , xi−1} for all σ ∈ G and for all i = 1, . . . , n. In particular,

we note that x1 is invariant. We set R[m] = F[x1, . . . , xm] for 0 ≤ m ≤ n

subject to the convention that R[0] = F. Then G acts on R[m]. For any non-zero f ∈ R[m], we may express f as a polynomial in xm and write

f = f0+ f1xm+ . . . + fdxmd

with fi ∈ R[m − 1] for all i = 0, 1, . . . , d where fd 6= 0 and we set degxm(f ) =

degm(f ) = d. We denote the leading coefficient fd ∈ R[m − 1] of f by c(f ).

Writing σ(xm) = xm+ αm−1xm−1+ . . . + α1x1, we have α(f ) =Pdi=0σ(fi)(xm+

αm−1xm−1+ . . . + α1x1)i. Therefore, σ(c(f )) = c(σf ) for all σ ∈ G. In particular,

if f is invariant, so is c(f ) since σ(c(f )) = c(σf ) = c(f ).

For each m with 1 ≤ m ≤ n, let φm ∈ R[m]G denote a fixed homogeneous

invariant with the smallest positive degree in xm among all invariants in R[m]G.

The existence of φm follows from the fact that the set R[m]G\R[m − 1] is

non-empty since N (xm) :=Qσ∈Gσ(xm) = x |G|

m + {terms of lower degree in xm} lies in

it. We take φ1 = x1. The invariants cm = c(φm) ∈ R[m − 1] will play a special

role.

Finally, note that we can make no claim as to the total degree of φm, in

particular, we cannot claim that the total degree of φm is less than or equal to

|G| for all m.

We first prove two lemmas.

Lemma 3.1.1. For any f ∈ R[m]G, there exists an integer k ≥ 0 such that

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Proof. We use induction on degxm(f ). When degxm(f ) = 0, there is nothing to

prove. So we may assume degxm(f ) = d > 0.

In the ring R[m](cm), the localization of R[m] at the multiplicative set

{1, cm, c2m, . . .}, the element φm/cm is monic as a polynomial in xm. Hence we may

divide f by φm/cm in order to obtain f = q0(φm/cm) + r0 where q0, r0 ∈ R[m](cm)

with degxm(r0) < degxm(φm). Thus

f = σ(f ) = σ(q0)(φm/cm) + σ(r0) = q0(φm/cm) + r0

for all σ ∈ G. Since

degxm(σ(r0)) = degxm(r0) < degxm(φm),

we see by the uniqueness of remainders that r0 = σ(r0) and hence q0 = σ(q0) for all σ ∈ G. Therefore q0, r0 ∈ R[m]G

(cm). Multiplying by a suitable power of cm, we see

that there exists an integer s ≥ 0 and polynomials q = cs−1 m q

0, r = cs mr

0 ∈ R[m]G

such that csmf = qφm+ r where degxm(r) = degxm(r

0) < deg

xm(φm). Therefore,

r ∈ R[m − 1]G because φm has the least positive degree in xm inside R[m]G. Since

degxm(q) = degxm(f ) − degxm(φm), by the induction hypothesis, ctmq ∈ R[m −

1]G

m] for some t ≥ 0. Therefore, for k = s + t we have ckmf ∈ R[m − 1]G[φm],

as required.

We note that it follows immediately from Lemma 3.1.1 that if cm = 1 for all

m, then any f ∈ R[m]G lies in F[φ

1, . . . , φm] as easily seen by induction on m. So

we have the following.

Corollary 3.1.2. If cm = 1 for all m = 1, 2, . . . , n, then

F[V ]G = F[φ1, . . . , φm]

is a polynomial ring.

Lemma 3.1.3. For any finite number of invariants h1, . . . , ht ∈ R[m]G, there

exists a monomial c = ck1

1 . . . ckmm in c1, . . . , cm, such that chi ∈ F[φ1, . . . , φm] for

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Proof. We use induction on m. First let m = 1. Since φ1 = x1 and c1 = 1,

the lemma follows from the previous corollary. Now assume m > 1. By Lemma 3.1.1, there exists an integer s ≥ 0 such that cs

mhi ∈ R[m − 1]G[φm] for all i.

Let cs

mhi =Pjajiφjm, where aji ∈ R[m − 1]G. Now, since the finite set {aji} ⊂

R[m − 1]G, by the induction hypothesis, there exist k1, . . . , km−1 ≥ 0 such that

ck1

1 . . . c km−1

m−1aji ∈ F[φ1, . . . , φm−1] for all i and j. Hence, for c = ck11. . . c km−1

m−1csm, we

have that chi ∈ F[φ1, . . . , φm] for all i = 1, 2, . . . , t, as required.

The following theorem shows that for a p-group, the invariant field is purely transcendental.

Theorem 3.1.4. Let G ⊆ U (V ) ⊂ GL(V ) be a p-group. Choose any set of ho-mogeneous invariants φ1, . . . , φn with the property that φm ∈ R[m]G is of smallest

positive degree in xm for 1 ≤ m ≤ n. Then F(V )G = F(φ1, . . . , φn). Furthermore,

there exists a non-zero f ∈ F[φ1, . . . , φn] such that F[V ]G(f ) = F[φ1, . . . , φn](f ).

Proof. We use the above notation. For the first part of the theorem, we need only to show that any h ∈ F[V ]G lies in F(φ

1, . . . , φn). (Since F(V )G =

{fi

gi |fi, gi ∈ F[V ]

G}, if f

i and gi lies in F(φ1, . . . , φn) then fgi

i does too).

As-sume h ∈ R[m]G\R[m − 1]. By Lemma 3.1.1, there exists an integer s ≥ 0 such that csmh ∈ R[m − 1]G[φm]. We write csmh =

P

kakφkm, where ak ∈ R[m − 1]G.

By Lemma 3.1.3, there exists some monomial cK = ck1

1 . . . c km−1

m−1 with cK · csm ∈

F[φ1, . . . , φm−1] (cm ∈ R[m − 1]G) and cK · ak ∈ F[φ1, . . . , φm−1] for all k. Thus,

h ∈ F(φ1, . . . , φm) ⊆ F(φ1, . . . , φn).

For the proof of the second part, let F[V ]G = F[a

1, . . . , al]. Since F[V ]G ⊆

F(φ1, . . . , φn) (F[V ]G ,→ F(V )G = F(φ1, . . . , φn)), we can write every ai as

hi/ki where hi, ki ∈ F[φ1, . . . , φn]. Multiplying with suitable polynomials in

F[φ1, . . . , φn] we can write common denominator instead of ki’s. Thus we can

write ai = hi/f , where hi, f ∈ F[φ1, . . . , φn]. Then

F[V ]G = F[h1 f , . . . , hl f ] ⊆ F[φ1, . . . , φn](f ). Therefore, we have F[V ]G ⊆ F[φ 1, . . . , φn](f ) and so F[V ]G(f ) ⊆ F[φ1, . . . , φn](f ) as required.

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The inverse is clear. Since F[φ1, . . . , φn] ⊆ F[V ]G we also have

F[φ1, . . . , φn](f ) ⊆ F[V ]G(f ).

3.2

Cyclic Group of Order p

We denote by Cp the cyclic group of order p with generator σ. There are p

non-isomorphic indecomposable representations of Cp in characteristic p; one

in-decomposable representation, Vr, of dimension r for 1 ≤ r ≤ p. And for the

dual space Vr∗ = homF(Vr, F), there is a basis {x1, . . . , xr} with the property that

σ(xi) = xi+ xi−1, setting x0 = 0. Therefore, the matrix of σ with respect to this

basis is σ =              1 1 0 . . . 0 0 0 1 1 . .. 0 0 0 0 1 . .. 0 0 .. . ... ... . .. ... ... 0 0 0 . .. 1 1 0 0 0 . . . 0 1             

Of course, any representation of V can be written as k1V1⊕ . . . ⊕ kpVp, and if

we choose a basis for V∗

{xi,j,r | 1 ≤ i ≤ r, 1 ≤ j ≤ kr, 1 ≤ r ≤ p}

with the property that σ(xi,j,r) = xi,j,r + xi−1,j,r, subject to the convention that

x0,j,r = 0, then we can identify F[V ] with the polynomial algebra

F[xi,j,r | 1 ≤ i ≤ r, 1 ≤ j ≤ kr, 1 ≤ r ≤ p].

Let us denote by F[V ]d the vector space spanned by the homogeneous

polyno-mials of degreed and note F[V ] = ⊕d=0F[V ]d. We have F[V ]0 = F and F[V ]1 = V∗,

and each element σi of Cp acts as a degree-preserving algebra automorphism of

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We note that F[V ]Cp = F[V1] ⊗ F[V 0 ]Cp, where V0 = (k1− 1)V1 ⊕ k2V2. . . ⊕ kpVp,

and so we may as well assume k1 = 0, that is, the representation V is reduced.

3.2.1

Construction of Invariants

Given any element f ∈ F[V ], we know that the trace of f , T r(f ) =Pp−1

i=0 σ i(f ),

and the norm of f , N (f ) = Qp−1

i=0σi(f ), are invariant. For example, in F[V2],

straightforward calculations show that N (x2) = xp2 − x p−1

1 x2 and T r(xi2) = 0 if

1 ≤ i < p − 1 with T r(xp−12 ) = −xp−11 . We note that N (x2) has degree p in x2

and since F[V2]Cp = F[x1, N (x2)], we may choose N (x2) as φ2 in the notation of

section 3.1.

Invariants in F[V ]Cp such as

s2 = x22− x1x2− 2x1x3

are invariant for all primes p. Invariants with this property are called rational invariants.

Shank [3] proposed an algorithm which constructs rational invariants with given (allowable) lead terms, for example, xm−12 :

s2 = x22− x1x2+ x1(−2x3),

s3 = x22(x2) + x1x2(−3x3) + x21(−x2+ 3x4),

s4 = x32(x2) + x1x22(−4x3) + x21x2(2x3− 6x4) + x31(−x2− 2x3− 6x4− 8x5).

In general [21, 3], using the graded reverse lexicographic monomial order with xm < xm+1, there is a rational invariant sm−1 with lead term xm−12 and with a

term (−1)m(m − 1)(m − 3)!xm−2

1 xm, and no terms involving xi for i > m. We

note that (m − 1)(m − 3)! is non-zero and hence invertible in F. Then in terms of the analysis of section 3.1, we may choose φm = sm−1.

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In the event, V has more than one non-trivial summand, say

Vi⊕ Vj ⊂ V

with i, j > 1, we need a particular kind of rational invariant, the two-dimensional determinant invariants. Denote the basis for Vi∗ by {x1, x2, . . . , xi} and for Vj∗

by {y1, y2, . . . , yj}. Then u = uVi,Vj = x1y2 − x2y1 is invariant, and there is

such an invariant for every pair of non-trivial summands in the indecomposable decomposition of V . We note that u has degree 1 in y2.

There is one more family of invariants that are needed, again associated to a representation with more than one non-trivial summand, say Vi⊕ Vj ⊂ V , with

bases as above and i ≤ j. Suppose f = f (x1, . . . , xi) ∈ F[Vi]Cp and consider the

ring F[V ][t] with Cp action given as usual on F[V ] and σ(t) = t. Now note that

f (x1+ ty1, . . . , xi+ tyi) ∈ F[V ][t]Cp

is invariant, since σ(xk+ tyk) = (xk+ tyk) + (xk−1+ tyk−1). But

f (x1+ ty1, . . . , xi+ tyi) =

X

k≥0

fk(x1, . . . , xi, y1, . . . , yi)tk,

from which it follows that fk = fk(x1, . . . , xi, y1, . . . , yi) is invariant. This process

of constructing new invariants with ‘mixed’ variables from existing invariants is called polarization, as we see in 2.3. We are interested only in applying this construction to the Shank invariants, and we want only the coefficient of t. This is the function f1 which has degree 1 in the y’s. In the three examples above,

these are (s2,i,j)1 = x2y1+ 2x2y2− 2x3y1+ x1(−y2− 2y3), (s3,i,j)1 = 6x4x1y1− 3x3x2y1− 3x3x1y2+ 3x22y2− 3x2x1y3− 2x2x1y1 + x21(−y2+ 3y4), (s4,i,j)1 = −24x5x21y1+ 16x4x2x1y1+ 8x4x21y2− 18x4x21y1− 4x3x22y1− 8x3x2x1y2 + 4x3x2x1y1+ 2x3x21y2− 6x3x21y1+ 4x32y2− 4x22x1y3 + 8x2x21y4+ 2x2x21y3− 3x2x21y1− x31(−y2− 2y3− 6y4+ 8y5).

Note that sk,i,j is defined for k ≤ min(i, j). Again, we note that the coefficient of

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3.2.2

Localized Invariants of Indecomposable

Representa-tions of C

p

We can compute generators for certain localized rings of Cp-invariants as follows.

First, let us consider the case V = Vn, and identify F[Vn] with F[x1, x2, . . . , xn]

such that σ(xi) = xi+ xi−1, with x0 = 0. We use Shank polynomials

sm−1 = xm−12 + . . . + αx m−2 1 xm,

for α non-zero in F. Recall that α = (−1)m(m − 1)(m − 3)!. As discussed in

section 3.1, we note that we may choose φ1 = x1, φ2 = N (x2) and φm = sm−1

for m ≥ 3 because each of these polynomials φm has degree 1 in xm. Further, we

note that for all m, cm is a power of x1.

Theorem 3.2.1. Let Vn denote an indecomposable representation of Cp, and let

φm be chosen as above and write x1 = x.Then

F(Vn)Cp = F(φ1, . . . , φn).

Furthermore,

F[Vn] Cp

(x)= F[φ1, . . . , φn](x).

3.2.3

Localized C

p

-invariants in general

We have that any representation of V can be written as k1V1 ⊕ . . . ⊕ kpVp and

that we may assume k1 = 0 as we see in 3.2.

We choose the largest m for which km > 0 and fix a particular copy Vl,m

of Vm. Let {x1, x2, . . . , xm} denote the usual basis for Vl,m∗ . Set x = x1,l,m and

x2 = x2,l,m. Note that σ(x) = x and σ(x2) = x2+ x.

For each other distinct non-trivial summand, say the jth copy of Vr, we have

r ≤ m, and we choose the usual basis {x1,j,r, . . . , xr,j,r} for Vj,r∗ , that is, we have

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We consider the polarized Shank invariants (si−1,r,m)1 for 3 ≤ i ≤ r, each of

which has a term xi−2x

i,j,r, which occurs with non-zero coefficient in F. That is,

each of these invariants has degree 1 in xi,j,r for 3 ≤ i ≤ r.

We also consider the determinant invariant

ur,m,j = xx2,j,r− x2x1,j,r

of degree 1 in x2,j,r.

Therefore, we may choose φ1,j,r = x1,j,r, φ2,j,r = ur,m,j and φi,j,r = (si−1,r,m)1

for 3 ≤ i ≤ r and we obtain Theorem 3.2.2.

Theorem 3.2.2. Let V = k2V2 ⊕ . . . ⊕ kmVm be any reduced representation of

Cp, where m ≤ p and km > 0. Choose a basis {xi,j,r} for the jth copy of Vr,

and identify F[V ] with F[xi,j,r | 1 ≤ i ≤ r, 1 ≤ j ≤ kr, 2 ≤ r ≤ m]. Fix a choice

x = x1,l,m and x2 = x2,l,m for some l and choose φi,j,r as above. Then

F(V )Cp = F(φi,j,r | 1 ≤ i ≤ r, 1 ≤ j ≤ kr, 2 ≤ r ≤ m),

and

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Bibliography

[1] P. Fleischmann, “The Noether bound in invariant theory of finite groups,” Adv. Math., vol. 156, no. 1, pp. 23–32, 2000.

[2] J. Fogarty, “On Noether’s bound for polynomial invariants of a finite group,” Electron. Res. Announc. Amer. Math. Soc., vol. 7, pp. 5–7, 2001.

[3] R. J. Shank, “S.A.G.B.I. bases for rings of formal modular semiinvariants [semi-invariants],” Comment. Math. Helv., vol. 73, pp. 548–565, 1998. [4] H. E. A. Campbell, R. J. Shank, and D. L. Wehlau, “Vector invariants for the

two-dimensional modular representation of a cyclic group of prime order,” Adv. Math., vol. 225, no. 2, pp. 1069–1094, 2010.

[5] M. D. Neusel, “Degree bounds-an invitation to postmodern invariant the-ory,” Topology Appl., vol. 154, no. 4, pp. 792–814, 2007.

[6] P. Fleischmann, M. Sezer, R. J. Shank, and C. F. Woodcock, “The Noether numbers for cyclic groups of prime order,” Adv. Math., vol. 207, no. 1, pp. 149–155, 2006.

[7] H. E. A. Campbell, B. Fodden, and D. L. Wehlau, “Invariants of the diagonal Cp action on V3,” J. Algebra, vol. 303, no. 2, pp. 501–513, 2006.

[8] R. J. Shank and D. L. Wehlau, “Computing modular invariants of p-groups,” J.Symbolic Comput., vol. 34, no. 5, pp. 307–327, 2002.

[9] R. J. Shank and D. L. Wehlau, “Noether numbers for subrepresentations of cyclic groups of prime order,” Bull. London Math. Soc., vol. 34, no. 4, pp. 438–450, 2002.

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[10] H. E. A. Campbell and J. Chuai, “Invariant fields and localized invariant rings of p-groups,” Quart. J. Math., vol. 34, pp. 151–157, 2007.

[11] H. E. A. Campbell and D. L. Wehlau, Modular Invariant Theory. Ency-clopaedia of Mathematical Sciences, 139. Springer-Verlag, Berlin., 2011.

[12] D. J. Benson, Modular Invariant Theory. London Mathematical Society Lecture Note Series, 190. Cambridge University Press, Cambridge, 1993.

[13] H. Derksen and G. Kemper, Computational invariant theory. Encyclopaedia of Mathematical Sciences, 130. Springer-Verlag, Berlin., 2002.

[14] M. D. Neusel, Invariant theory. Student Mathematical Library, 36. American Mathematical Society, Providence, RI., 2007.

[15] M. D. Neusel and L. Smith, Invariant theory of finite groups. Mathematical Surveys and Monographs, 94. American Mathematical Society, Providence, RI., 2002.

[16] L. Smith, “Polynomial invariants of finite groups. a survey of recent devel-opments,” Bull. Amer. Math. Soc. (N.S.), vol. 34, no. 3, pp. 211–250, 1997.

[17] H. E. A. Campbell and I. P. Hughes, “Vector invariants of u2(Fp): a proof

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[21] H. E. A. Campbell, “Rings of invariants of representations of cp in

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